Paper 2, Section II, B

Vector Calculus | Part IA, 2020

Write down Stokes' theorem for a vector field A(x)\mathbf{A}(\mathbf{x}) on R3\mathbb{R}^{3}.

Let the surface SS be the part of the inverted paraboloid

z=5x2y2,1<z<4z=5-x^{2}-y^{2}, \quad 1<z<4

and the vector field A(x)=(3y,xz,yz2)\mathbf{A}(\mathbf{x})=\left(3 y,-x z, y z^{2}\right).

(a) Sketch the surface SS and directly calculate I=S(×A)dSI=\int_{S}(\nabla \times \mathbf{A}) \cdot d \mathbf{S}.

(b) Now calculate II a different way by using Stokes' theorem.

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